WEBVTT
00:00:00.886 --> 00:00:04.971
PROFESSOR: How about
the expectation value
00:00:04.971 --> 00:00:09.329
of the Hamiltonian in
a stationary state?
00:00:09.329 --> 00:00:11.370
You would imagine,
somehow it has
00:00:11.370 --> 00:00:14.520
to do with energy ion
states and energy.
00:00:14.520 --> 00:00:17.370
So let's see what happens.
00:00:17.370 --> 00:00:20.100
The expectation value
of the Hamiltonian
00:00:20.100 --> 00:00:23.610
on this stationary state.
00:00:23.610 --> 00:00:31.056
That would be integral
dx stationary state
00:00:31.056 --> 00:00:35.980
Hamiltonian stationary state.
00:00:39.452 --> 00:00:42.630
And we're going to see
this statement that we made
00:00:42.630 --> 00:00:45.146
a few minutes ago become clear.
00:00:45.146 --> 00:00:50.505
Well what do we get
here? dx psi star
00:00:50.505 --> 00:01:00.415
of x e to the i Et over
h bar H e to the minus i
00:01:00.415 --> 00:01:06.394
Et over h bar psi of x.
00:01:06.394 --> 00:01:13.290
And H hat couldn't care less
about the time dependence,
00:01:13.290 --> 00:01:16.820
that exponential is
irrelevant to H hat.
00:01:16.820 --> 00:01:21.630
That exponential of time can be
moved across and cancelled with
00:01:21.630 --> 00:01:22.150
this one.
00:01:24.720 --> 00:01:27.555
And therefore you
get that this is
00:01:27.555 --> 00:01:37.898
equal to dx psi star of x H hat
psi of x, which is a nice thing
00:01:37.898 --> 00:01:39.722
to notice.
00:01:39.722 --> 00:01:44.800
The expectation value of H
on the full stationary state
00:01:44.800 --> 00:01:48.050
is equal to the
expectation value of H
00:01:48.050 --> 00:01:52.816
on the spatial part of
the stationary state.
00:01:52.816 --> 00:01:54.760
That's neat.
00:01:54.760 --> 00:01:56.802
I think it should be noted.
00:01:56.802 --> 00:02:03.350
So it's equal to the
H of little psi of x.
00:02:03.350 --> 00:02:06.970
But this one, we can
evaluate, because if we
00:02:06.970 --> 00:02:13.884
are in a stationary state, H hat
psi of x is E times psi of x.
00:02:13.884 --> 00:02:17.250
So we get an E
integral the x psi
00:02:17.250 --> 00:02:21.340
star of psi, which
we already show
00:02:21.340 --> 00:02:25.950
that integral is equal to
one, so we get the energy.
00:02:32.950 --> 00:02:35.280
So two interesting things.
00:02:35.280 --> 00:02:37.950
The expectation value
of this quantity
00:02:37.950 --> 00:02:40.590
of H in the stationary
state is the same
00:02:40.590 --> 00:02:44.110
as it's quotation value
of H in the spatial part,
00:02:44.110 --> 00:02:47.724
and it's manually
equal to the energy.
00:02:50.390 --> 00:02:59.700
By the way, you know, these
states are energy eigenstates,
00:02:59.700 --> 00:03:06.090
these psi of x's, so you
would expect zero uncertainty
00:03:06.090 --> 00:03:09.120
because they are
energy eigenstates.
00:03:09.120 --> 00:03:14.815
So the zero uncertainty of the
energy operator in an energy
00:03:14.815 --> 00:03:15.757
eigenstate.
00:03:15.757 --> 00:03:20.460
There's zero uncertainty even
in the whole stationary state.
00:03:20.460 --> 00:03:28.705
If you have an H squared here,
it would give you an E squared,
00:03:28.705 --> 00:03:32.860
and the expectation
value of H is equal to E,
00:03:32.860 --> 00:03:36.610
so the expectation value of H
squared minus the expectation
00:03:36.610 --> 00:03:40.480
value of H squared
would be zero.
00:03:40.480 --> 00:03:43.595
Each one would be
equal to E squared.
00:03:43.595 --> 00:03:47.780
Nothing would happen, no
uncertainties whatsoever.
00:03:47.780 --> 00:03:55.820
So let me say once
more, in general, being
00:03:55.820 --> 00:04:02.298
so important here is the
comment that the expectation
00:04:02.298 --> 00:04:18.270
value of any time independent
operator, so comments 1,
00:04:18.270 --> 00:04:30.965
the expectation value of any
time-independent operator
00:04:30.965 --> 00:04:44.000
Q in a stationary state
is time-independent.
00:04:47.840 --> 00:04:49.646
So how does that go?
00:04:49.646 --> 00:04:51.062
It's the same thing.
00:04:51.062 --> 00:04:56.600
Q hat on the psi of
x and t is general,
00:04:56.600 --> 00:05:07.069
now it's integral dx capital Psi
of x and t Q hat psi of x and t
00:05:07.069 --> 00:05:12.162
equals integral dx-- you have to
start breaking the things now.
00:05:12.162 --> 00:05:18.206
Little psi star of x E
to the i et over H bar.
00:05:18.206 --> 00:05:20.540
And I'll put the
whole thing here.
00:05:20.540 --> 00:05:27.910
Q hat Psi of x E to
the minus i et over H.
00:05:27.910 --> 00:05:32.330
So it's the same thing.
00:05:32.330 --> 00:05:40.700
Q doesn't care about time So
this factor just moves across
00:05:40.700 --> 00:05:44.904
and cancels this factor.
00:05:44.904 --> 00:05:47.700
The time dependence
completely disappears.
00:05:47.700 --> 00:05:51.843
And in this case,
we just get-- this
00:05:51.843 --> 00:05:59.702
is equal to integral dx psi star
Q psi, which is the expectation
00:05:59.702 --> 00:06:07.270
value of Q on little psi
of x, which is clearly
00:06:07.270 --> 00:06:11.730
time-independent, because
the state has no time anymore
00:06:11.730 --> 00:06:14.900
and the operator has no time.
00:06:14.900 --> 00:06:22.180
So everybody loves their
time and we're in good shape.
00:06:22.180 --> 00:06:24.792
The second problem is
kind of a peculiarity,
00:06:24.792 --> 00:06:30.506
but it's important to
emphasize superposition.
00:06:30.506 --> 00:06:33.910
It's always true,
but the superposition
00:06:33.910 --> 00:06:40.940
of two stationary states is
or is not a stationary state?
00:06:40.940 --> 00:06:41.844
STUDENT: No.
00:06:41.844 --> 00:06:43.080
PROFESSOR: No, good.
00:06:43.080 --> 00:06:46.720
It's not a stationary
state in general
00:06:46.720 --> 00:06:49.110
because it's not factorizing.
00:06:49.110 --> 00:06:54.010
You have two stationary states
with different energies,
00:06:54.010 --> 00:06:58.408
each one has its
own exponential,
00:06:58.408 --> 00:07:02.510
and therefore, the whole
state is not factorized
00:07:02.510 --> 00:07:05.050
between space and time.
00:07:05.050 --> 00:07:08.328
One time-dependence has one
space-dependence plus another
00:07:08.328 --> 00:07:10.202
time-dependence and
another space-dependence,
00:07:10.202 --> 00:07:13.000
you cannot factor it.
00:07:13.000 --> 00:07:16.740
So it's not just a plain fact.
00:07:16.740 --> 00:07:29.446
So the superposition of
two stationary states
00:07:29.446 --> 00:07:39.924
of different energy
is not stationary.
00:07:46.307 --> 00:07:51.310
And it's more than just saying,
OK, it's not stationary.
00:07:51.310 --> 00:07:54.310
What it means is that if
you take the expectation
00:07:54.310 --> 00:08:00.690
value of a
time-independent operator,
00:08:00.690 --> 00:08:04.740
it may have time-dependence,
because you are not anymore
00:08:04.740 --> 00:08:07.222
guaranteed by the
stationary state
00:08:07.222 --> 00:08:10.630
that the expectation value
has no time-dependence.
00:08:10.630 --> 00:08:16.850
That's how, eventually, these
things have time-dependence,
00:08:16.850 --> 00:08:19.620
because these things
are not [INAUDIBLE]
00:08:19.620 --> 00:08:21.250
on stationary states.
00:08:21.250 --> 00:08:23.302
On stationary
states, these things
00:08:23.302 --> 00:08:27.694
would have no time-dependence.
00:08:27.694 --> 00:08:31.500
And that's important,
because it would
00:08:31.500 --> 00:08:33.740
be very boring,
quantum mechanics,
00:08:33.740 --> 00:08:36.429
if expectation
values of operators
00:08:36.429 --> 00:08:38.419
were always time-independent.
00:08:38.419 --> 00:08:39.515
So what's happening?
00:08:39.515 --> 00:08:42.616
Whatever you measure never
changes, nothing moves,
00:08:42.616 --> 00:08:44.590
nothing changes.
00:08:44.590 --> 00:08:47.500
And the way it's
solved is because you
00:08:47.500 --> 00:08:50.100
do have those stationary
states that will give you
00:08:50.100 --> 00:08:51.674
lots of solutions.
00:08:51.674 --> 00:08:53.790
And then we combine them.
00:08:53.790 --> 00:08:56.640
And as we combine them,
we can get time-dependence
00:08:56.640 --> 00:09:00.290
and we can get the most
[INAUDIBLE] equation.